The Laplace T,'ansform of the Bessel Functions

where n ~ 1, 2, 3~ ...

by F. M. RAhaB (a Cairo, Egitto - U.A.R.)

(x) oo Summary. - The integrals / e--PtI(~ (1)xt~-n tl~--~dt and f e--pttk--iJ f 2a~t=~n i 1] di, n=l, 2, 3, ..

o o are evaluated in term of MacRoberts' E--Functious and iu terms of generalized hyper. geometric functions. By talcing k~l, the of the four functions in the title is obtained.

§ 1. Introduction. - Nothing is known in the literature ([1] and [2]) of the Laplace transform of the functions

where n is any posit.ire , except the case where the argument of the Bessel functions is equal to [~cti/2] (see [1] p. 185 and p. 199). In this paper the following four formulae, which give the Laplace transform of the above functions, will be established: / co 1 (1) e-Pttk--iKp xt dt=p--k2~--~(2u)~-'~x ~

o

](2n)2"p2]~_k nk ~ h n; : e :ei~ [~4x-~--]2 E A n; ~ + 2 , 2 2: (2n)~"p ~

1 [[27~2n~2~ i . 1

E ~ n; 2 ~- , A n; 2 : 2: (2n)2"p 2 e:~ where R(k + U) ~ o, R{pI ~ o and n is any positive integer. The symbol h(n; a) represents the set of 318 F.M. RAGAB: The Laptace Tra.ns]orm o~ the Bessel Functions, etc. parameters o~ a+l a+n--1

8 (2~ e-ptt~-I xt dt = 2 k-~ r:

o [ 1 1 1 ( 1 ) ( ~, 1 I, k, k+ h n; ~ h n ~, -~i ~ 2 )' ~ , ; --~. ~ ::4,~+:n2. e~ where R(k) > o, R(p) > o and n is any positive integer. The symbol A(n;~) has the same meaning as in (1) and the symbol Z means that to the expression following it a similar expression will --i instead of i is to be added ;

oO 2V ~ 2Y 1 (3)

0

X=o k! l~(2v --{- ~, + 1) [n'nP" l

k v+). k+l v+~. ~i.= 4a" , -- " e n2op 2 2+ n 2 + n ' -- ~F2.-: X+l k+n 5(n; 2v+l) where

and n is any positive integer. The symbol A h~s the same meaning and

• t . the asterisk denotes that the factor -m the set of parameters n

X+I ),+2 X+n n n

is omitted;

I] v k --v---1 dt --p --k+-9,.~

0 F. M. RAGAB: The Laplace Transform of the Bessel Fu, nctio~s, etc. 319

I p~ l~_~_ ~ +~k

oF2,~+~ ; A n; 1+ --v A; n; 1-~ nk + v 2 ' 2 ' 2 ; 4(nx) 2ne±~n~

nk -t- n -k v] 3 p~e -~'~] oF~.+l[; ~(n;1 + .k +~2 -~)' A(,~; 1 + 2 ] '2;~J

Y --V -}- p-~+ ~ m

(__ 1)z2,Tn~z n I p2 iz_ z! v(v +)` + 1) ~"

Y, ),+1 ),+n 1--k }_v-~2), 2--k v-~2)` ' oF2~+1 ; n n ' ~ - , ~ + - ,I

"Pze~=inrc ] _ A(n; 1 + v + )`) : 4(nx)~ . where R (k--~ v) >o, R(p)>o and x is taken for simplicity to be real and positive. The function appearing on the right hand sides of (1)and (2) is MAcRoBER~'S. E function whose definitions and properties are to be found in (3} pp. 348-358. The symbol A and the asterisk have the previous meanings. These four formulae will be proved in § 3 by means of sub- sidiary theorems which will be stated an proved in § 2. In § 4 some known results and other new results will be derived as particular cases. The following formulae will be required in the proofs. ([4] p. 92, form. (7).) :

(5~ f e-)')`~-lE ; ~,, : q ; ~ : ~ o) d), ----- 2 k-17:- 1 0 320 F. 5{. RAGAB: The Laplace Transform of the Bessel F,t~nctions, etc.

1 1 ~ p p i i i ] ~q (2~) ~-~" <~+P-q)+"-~ '~ ~- ~ P~÷ -~P-~

n-- i l • E (-- 1)x[4z-~n-"(~+q-p)];

k ~ k-lrl t 1 ,, p; h(n; a,.+X), 2+n 2 +n:4 z n"(~+q-V)e:~('+~)~ E ~.+--, 1 ...... , --,X+n q; A(n; p~+~) n Tt

where R(k)> o, p ~> q + 1 and n is any positive integer. For other values of p and q the formula holds provided that the integral is convergent. ([3], p. 352): if p--~q, then

(6) E(p; a,.:q; p,:z)= F(p~)r(~)...]:(av) ... F(p~)~F" (p ; a,.: q; ~; _~zl) .

([3], p. 353): if p_>.q+l, then

E(p ; ~ : q ; p~ : z) = Z; IIP' P (a, -- a,. )l lIv r(~t-- a,.) I-~ r(~Az%

F (a,., a,.-p,+l, ..., a,.-- gq -}- l; (--1)p-qz) (7) \a,. -- a~ + l, ... ~ , a,.--a~+l

([3], p. 409): if p~q+l, then

p E(p; a,.: q; p~ l z)-~--uP-q-lE [I sin (pt--a,.)u

a,., ~ -- ;~ + ;q + 1: e~:~:(v-q-1)z t (8) I 1~' sin (a s-- ~,-)r:l-lE ( :or--al+ 1, ... * ..., a,,-- a~+l

([3], p. 411):

(9) E(: v+l: z)=z~J~

(13), p. 154, ey. 5): F. ~. RAGAB: The Laplace Tra~tsform o] the Bessel F~a~wtion.s~ etc. 321

n-- 1 1 1 1

Also the followi.ng formulae are required:

(11) r(z)t'(t -- z)= r~/sin ~z;

k~ (k + 1)~ (k + n -- 1)7: (12) sin -- sin ... sin --- 21-n sin kr~ Tt ~ n

(13) sin sin -- ... sin -- -- 21-'n Tt n ~

For the derivation of the particular cases, use will be made of the following formulae :

F 2, =Vv i+ Tvc )

(14) can be obtained by differentiating the expansion

1 {l+Vl--zt--2b (15) F(b, b+ ~ ; 2b + 1; z) "-- t 2 t which can be extablished by means of LAC~nA~GE expansion, and then 1 putting b=~--l. ([5], p. 103, ex. (38)):

(16) c(1--z)F(a, b; c; z) -- oF(a --1, b; c; z)

------(o--b)~F(a, b; c+ l; z)

([3]; p. 305);

(17) P~'-v(z) "-- F(v + 1) ze-v~- ' 2 ' where P,-~(z) is the associated LEG]~I,~Dm~ finetion of the first kind. And the known transformation

(18) F(a, b ; c; z) -- (1 -- z)C-'~-~'F(c -- a, o -- b; c; z)

Annali dl Matematica 41 322 F. ~. RAGAB: The Laplace Tra~sform of the Bessel Fu~wtio~s, etc.

§ 2. - Subsidiary formulae - The thorems to be used are: ([6]; p. 304):

1 1 1 ~.\ i 9) K~,(~) = U=1 ~, E_~ f1E 1,~, 2g,::4me )

1 (20) x -~ Z :E(p; ~+5~, ]: q; 9~+},: d~z) -- i, --i 1 E -:E(p; a,,, I:q; ;~: d "~z) i,--i

(20) can be proved by expending each E--function on the left by means of (7) and combining the two resulting expressions by factoring out common terms applying (11).

Ifp~q+l, lampz[ ~, Rk+-~-/( 2art>o, r:l, 2, 3, ..., p and n is any positive integer, then

Oo -(91) e-lXk-~E p; oq~: q; ~,: zX~ dy= o

i i p q I T i 1 - ~ ~+--~." + ~ (2.)(~-~')(~+~-°)~2 -~ ~'-~ ~'- ~-~+~.

• k,., ~ (~) --/~;+~, v, ~(z)

i

where

1 (21') ~, p, q (z) ---~ cosec ~z ~k .

.E A n; 1 +-~), .~, q; ~ '*k/

and 3 1 Rk +~. p, q (z) is k, .. q(z) F. ~¢[. RAGAB: The Laplace Transform of the Bessel Functio~s, etc. 323

3 i with (k + 1) instead of k and 2 instead of the 2 which appears in the last line of (21'). Also U"~, p, q(z) is given by

~, ~, q(z) = (-- 1) x cosec ~ -- n 7: cosec 2 7:"

E ),.+1 k+n 1 k ), l k ), ~'"'*'"'~' 2 2+~-,' ~ + n, q; h(n; ~+k)

Io (2[) the symbol h and the asterisk have the previous meanings. For other values of p and g (21) holds provided that the integral is convergent. Iv prove (21) consider the special case with p ~ 1 and q-- 0, a~= :¢, then the left side becomes co Go 2 z ~ e-Xk~:+~--lE a

0 0 2~ 1 ~--1 ~. -- z~F2 ~+ ~--~-~n ~-~ Z (-- 1)~(4z'*); ~.= 0

• * -- : - e=~(~+l) ~ _by 5. n ' 2 ~- n n n 4 z~

Now expand the last E-funtion by means of (8) taking in (8} a~ =

= k a ), k+l ---:¢+)' and o~ t+~--no: +~k+t (t--0, 1, 2,..., n--l), ~ + ~ + n' ~ - 2 -~ n then the last expression becomes CG -~X~-IE ~ :: zX ~ d), -- z~2 k +~-1 ~?-n~-i

o

(22) ~, (-- 1)~(4z')~ A k Ak+~,~'~ x(z) ).------0

2_~ 1 3 ¢$ -- 1 ~--I + z~2 k+ ,~ r:~n~-~ ~ v, Bx", t(z). where

sin + ~ t e:~i~(n+~)~-" ~ + A ~' ~(z) = It=o n 2 n n -- 1 '*

sic(1 1sin -- +~s)l ~ sin , 2 n s=o 324 F.M. R~GAu: The Laplace Truns]orm of the Bessel Fur~ctio~s, etc.

k ik .+~ k ~--1 , - : e-~(n+l)ni4Zn 2+ n '2 4 n r-~, ... ~+ n E ~1 k 1 k I ~, l+~,'"*'",nq- ~

8 3 and A"'~ (z) is the value of A~',~(z) with k -+- 1 instead of k and ~ instead k+l,~. of the ].1 Also B"~,t(z) is given by ...

B~", t(~) = (-- 1)t+:' t=o n n 4

sin I t+n ;~ +e~: .--IIsin--n --n ~sin 2

[ ~-]--- k -]- t, c¢ + t--1 +1,...*..., :¢ + t': e~=(,+~).i4z. n n E t k 1 t k t+l t+n J._ l*n 2' 2--n 2' n ' ' n Here apply (11) and (12) to substitute for each sine product in (22). The n~ 8_ asterisks in the three E-functions appearing in Ak;~(z), A;'g:,x(z) and Bz~,t(z) denote that the three parameters

k ~. ), k+l :¢ ), anda-i-~-l-t ~+~+~+' 2 +~+n n are omitted in each E-function respectively. Also the above set of para- meters in the last E-functions can be written, using the meaning of the symbol h for ),--0, 1' ..., n--1 in the forms

nk+l h(n; ~), h(rt;:~Lq-~) and h(n; 2 + ~) respectively. This makes each E-function independent of ~ because the lower set of parameters in each E function is independent of k. This n--1 enables us to take evory E-function as common factor autside the sign Z. k~0 So that by changing the arder of summation in the last double finite peries in (22), it becomes

oo e-~),k-lE :¢::z),7~ d),--- 2~"1~-~n~-1 sin -+-~

0 F. M. RAGAB: The Laplace Transform of the Bessel F,to~ctions, etc. 325

e -~(nq-1) I 2k aq-~. n R;' (z)~ (--1)~, ~,~,o ),=o sin 2-1-~ ~

e~i~fn+ 1) ~------d- n--1 I I k+~+~+~ Z (-- 1)~

[ , 1 ).+~+t ]

-q- 2h--lU ~n a-~ Y, sin (t q- a)u U"t, 1, o(z) E (-- 1) )...... L ~ t=o ~,=o sin (X +c q- t) ]

Now wrile e~:~ for e~( ~+~)~ when n is even and wrile 1 for e-~("+w~ when n is ~dd. Then sum each of the last three s~ries E by applying ~.=o the following two summations respectively

(23) E (--1) ~ ~ (n is an even -}- ve integer) ~:o sin (x+ ~)~ sin (nx) r:

and (-- 1)). n (24) (n is an odd q- ve integer) sin a~ +~ r: sin (nx)r:

Thus the last expression becomes exactly formula (2l) with p---1, q : O. The general case con be deduced in the usual way of generalizations.

§ 3. Proofs of the formulae (1) -(4). - Iv prove (1), apply (21) twice 1 ~ 1 with z -- 41 x2e~~ und z~ ~x-~ taking p~3, q=0 with al---~l, ~-~ ~ ~, 1 and :¢a-- -- ~ ~t. Then add the two resulting expressions making use of (19) It may be noted that the ternis orising from the series E U"~, p, q (z) in (21) cancel because eisx-- e-~sx--= (-- 1)x(k is any -}- ve integer). Thus (1) is proved When k--1, then (1) gives the LAPLACE transform of 326 F. ~. RAGAB: The Laplace Transform of the Bessel Functions, etc.

~ts

1 1 I (25) P - ~2-~(2~) ~-"~ L[(2~)~"P~]4~"1 ~

h n; , A n; : (2n)~,,p ~ E 1

(2n ~. --p ~2 (2=) i

4Xe.e~=i _

E 3

/ u. where R(p)~O, Rtl-t-~,J~0, ~ real and positive and n is any positive integral. 1 PROOF OF (2)- To prove (2) apply (5) twice with z~ 71 x2e~ and 1 z-~ ~ x~e -i~ respectively taking p--- 3, q -- 0 with

1 1 o:~--1, a~ lz and ~------~ t~, so getting if R(p)~ O, R(k)~ 0 and w is real and positive 0o I 1] 1 xt-~ dr-- u -~-(2T:) ~-" n p

0

)<--oL ~ P" I ~,--~ k ), k--t-1 X x~'p2p~d ~] 1, ~ -{- ~, 2 4- n :: 4~+~n TM ]

v E(1, k k-~l h n; h n; -- ::--d ~/ by (20) ~,-1 ~ 2' 2 ' ' 4'~+~n TM ] F. M. RAGAB: The Laplace Tra~sform of the Bessel Fu~wtions, etc. 327

Thus (2) is proved. When k = l, then the LA~'ZACE transform of

is rc-3/~(2~) ~'E 1E 1, 1, 1 ~. 2 (26) 4p ~,_~i ~,A n; , An;-- 2/..4~+~n2,, e ~ where R(p)~ O. It may be noted that when the E function appearing in (26) is exponded by means of (7) the first two series are non existent because a~-~-~2~-~-1. However these two series can be replaced by a certain limit (see (7), p. 30).

P]~ooF oF 3. - To prove (3), take in (5) p~0, q:] with 9~2v+1, wrile k-}---2v for k and c~-~ for z, apply (9) and so abitain (3). When k=l, then (3) gives the LAPLACe. transform of

a--~p~+ ~2-~ u~J~ 2a~tg in the form

(27) E (- ~)~,,~ r + r 1 +-~ / z ~ r(2~ + z + ~)I [ 4~- 1~ ~o l~-p~i ~

-n-' 1 +v- n -, n2,p 2

,...*...,)~ ~ n, h(n; 2v + 1) where

>0, R(p) > 2 i I and n is any positive integer.

Pnoo~ o~' 4: - In (21) take p~0, q----1 with ~ .~ v -~ 1, write k----2v for k and so obtain (4) by an application of (9).

§ 4. Particular cases. - We are now in a position to obtain a large number of particular cases of the formulae (1). (2!, (3) and (4). Some of these results are known (see [1], p. ~85 and p. 199) and the rest are new. To derive these results the following formulae are also required: 328 F.M. RAGAS: The Laplace Transform of the ,s, etc.

([8], p. 759): For all values of p and q E(p; n$~tr:q; m~s:ze+~)'--(2g) -~'(m-1)(p-¢-~)

(28) ~=o\ Z / I zme#i~ ] p; h(m; m~,.-}- n): m,.(v-¢-~) E /n'+l n L ~- '"'*"" +ram q; A(m;m~,+n) and ([3], p. 351, eq~. (15)): )(~ )'°(;; (p)

=E (~+ ~A--v, 2+~ ~--v:: ;)

1 Thus in (1) take ~=2v, x=2a ~/~, n~2 and k=~--}- ~ then it becomes oo t i i i fe-P t~-i K2,~(2ait-~)dt ___--p - ~-~ 2 ~-~(2r~)-i = ~~-~)

0 __ --v_t_l ~--vj_3 1 a~: e*'~) )] --g-+i' -E +~' ~2--~' -E -:t:2 4p ~ 5 t~--v, 3 ~--v 5 3 a ~ )( , p,~v+4, 2 tT~ ' 2 +4:2:4p ~e+~ /1 1 a) --2-1a-~-~Et~ + ~t--v, ~+ ~t + v::~ , by (28)

=~a-~p-~r1,(~) ~+v+ Y ~--v-4-2~)1° e~W_~,,~(~)p) by (29)

Thus we have known result (see [1], p. 199 form (39)) namely

c~ (30) ~_ ,)w ~,(;)

0 F. M. RAGAB: The Laplace T~'(~nsform of the Bessel Functioas, etc. 329

1 where R(~t-4-v)>--~, R(p)> 0. It may be noted that the formulae (32), (33), (34), (35) and (36) in ([1], p. 199) can easily be deduced from (30). Again in (1) take n~ 1 and get

(31)

O

2 ' 2 ; 2; ~ --2~-~P-~F k-~-~12 ~ k 2~-~1

2 ' 2 ;~;~

where R(k ~ ~)> O, R(p)> 0 and ~ is real and positive. Formula (31) may be compared with ([1] p. 198 formulae (25), (26), (27) and (28)). In (1) take k: 1 and n=3, so getting the LAt'L~CE transform of the function

K~ 03 , namely

] ( 1 ~ 5 ~ 7 ~ 1 ~ 5 7 ~ ~t''\-'-'-e~:'~/ 1 4036 1 E ~+~, ~)+6, ~)+~), 2 6'6 6' 6 6"2"6~p ~ ] (32) • / (63p) ( ~ 4 ~ 5 ~ ~ 4 ~t 5 ~ 3 4~c6 a. \ E 1+~,~+~, 3+~,,1--~,~ 6'3 ~26op ~ ] it. where R(p) > O, R( 1.4---3 1 ~) > 0 and x is real and positive. in (2) take k--1 and n ~-1, so getting the LaPLaCE transform of

namely if Rip) > 0

2-2rc-~p -1 Y. 1E 1, 1, 1 1 1 .032p2 (33) where a~ is real und positive. Thus is a new result.

Annali di Matematlca 42 330 F. M:. RAGAB: The Laplace Transform of the Bessel Fu~ctio~s, etc.

In (2), take k:l, n--2 and get the LAPLACE transform of

, namely

(34) 2-27:--~p ~,_~, -:E l, 1, 2' 4' 4~2' 4' +2:: 4 ~ e~ where R(p)>O. 1 In 26 take ~ =~ and so obtain-the LAPLACE transform of

namely

(35) 2-2r:-~(2n)~-'~p-~[~|~8 /2 ~\1 E 1 _ E 1, 1, 1(1) A n; \~:/ i,-.1 ~ 2' '

where R(p)~ 0 and x ~is any positive integer. 1 In (29) take n-----1 and write ~ v for v, so getting

e-P~J~ 2a~t dt~a~p-l-~F ~, 1-{-~;v~l;--

0

by (14).

a 2 Here write ~-for a and so obtain the L.APLACE transform of

(36) J~(at) samely r--l~ where

B(v) > -- 1, R(p)~ [I(a)[, r--(p*Wa~)~ and R--p-[-r.

(36) is a known result (see [1], p. 182 form. (1)). F. M. RAGAS: The Laplace Transform o] the Bessel Functions, etc. 331

1 1 2 Again in (3) take n--l, k=2, write~v for v, ~a for a and get if R(v) > -- 2

--PttJ~(at) dt 2a~rc ~p-~-~ v(1 + v) 0

• F 1+~,~+~; l+v;

~a~2~-~p -~-~ l+~vi l+p-~a~ "F + ~,1v +~; v l+v;-- a 2

+i--4~\~)F1+~, +~;2+~; by (16) " " I

a~p-~-~2-~(i -{- v) p~r -8 + • --v ,pSr_~(2~)~+~ I by (14) 2(1 -4- v) where r -- (p~" -Jr- aZ? i2 an4 R -=p -t" r. Simplifying, we get the known result ([1], p. 182, ex. (2)) namely

GO (37) f e-pttJ~(at)dt = r-3(p + vr)(a ] R?,

0 where R(v) > -- 2, R(p) > [ I(a) I, r -- (p2 + a~)i/z and R --p -{- r.

In (3) take n=l, k---re+l, 2v--m(m is any positive integer) and az write ~-for a, then it becomes form. (4) ([1], p. 182) namely

(38) /e-~ttmJm(at) dt --- 1.3.5 ... (2m -- 1) (~m~,-2m--1 ] o where R(v) > 0 and r --- (a 2 + p2)1/2. In (3) take k---,0 and n--i, and get

OD f e-Vtt-IJv(at) dt ----- 2-1n-~ F ,~+~;1 l+v;-- o;) r{l+v) 9

(a ~ -4- p~)tt~ + p by (15). ° t 332 F. ).[. RAGAB: The Laplace Transform of the Bessel Functio+,s, etc.

Thus we have obtained ex. (5), p. 182 of the reference [1] namely if R(v)>0 and R(p) >-- Jim(a)]

¢x) (39) / e-~t t-~ J~(at)dt _vii p + (a ~o + p~)lr~ ] 0 Also in (3) write a~/4 for a, v for 2v, then take n--1 and k-~--v~-I and so obtain ex. (7) p. 182 of the reference [1] namely / (DO 1 (40) e-ptt~J~(at)dt -- 2~u-~P v + a~(p ~ -]- ct~)-~-~

0 1 where (Rv)>--~ and R(p)~ lira(a)].

~b 2 Y Again in (3) write -~- for a, ~ for v, take k-~-bl and n-~l; so getting if R(t~ -+- v) ~ -':1- r( F+V+2 1) r(p'+V+2)2 l e pt tFJ~(at) dt -- 2'~a~p-~-~= - r(v + 1) 0

2 ' 2 ; v -[- 1 ; -- -- a~2~p-~-'~-~-~

a2) 1 a~\_F__~ _ (1 + v -- l-t v--p.; v+l;--~ by (18). ' 2

Here apply (][7) and so obtain ex. q. p. 182 reference [1] samely

(41) f e-~tt~J~ (at)dr __~ r(1-}- ~-{- v)r-~-~p~-~(P) ,

0 where

R(v --}- t~) >- -- 1, R(p) > I Ira(a) 1 and r -~ (p2 _+_ a2)112 1 In (3) take n ~ 2, k---I~--}- 2 and get after making use of (6)

co f e--~ tF-~. J~ 2a~ll t~ I di ---- a~ p-~-~-~21 ~+~" 11~

O F. M. RAGAB: The Laplace Transform of the BesseI Fu~wtions, etc. 333

3 ~+v+k 4p z \ ' 4 + 2 ; a ~ e=~" £ (--1)~(~]~E + --_ ).=o \ P/ + 1, ),+1 , i ~-., V--l-2,v + 1 )

~ a ~p-~-~-] E (; +l~+v:l+2v: :) by (28),

~+v; 1 + 2V;--p) by (6) =a~P-~-~-~ r(l+2v) ~F~(~+

Now substitute for 1F~ in terms of the known whittakor's function ~, ~(;), getting

J a (~0) (42) e-"Ptt ~- ~I(11) J~-~ 2a~ t] dt -~ a- ~1 p- ~ i~(2v+ 1) e- ~ M~, 0

1 where R(v+3)>-- ~and R(p)>O (42) is ex. (34) p. 185 of [1]. In (3) take n--3, k = 1 and get the L,~'LACm transform of

namely

2 1 2 2 (43) 2~-~a~p-~ ~-~ E (-- l);'35;`

v + ), 4a 3 1 + - -if- , 1 3 ; 729p ~ ~F~ ~+1 ),+3 2 1 2 2 2 ] 3 '3v+3'3v+~]'3 v "}- 1 where

Important particular cases are obtained from (3) when the REISEL 1 function Jz~ is changed to trignometric function. Thus, if in (3) k-- 1 + ~-~ 334 F. ~V[. RAGAB: The Laplace Transform o] the Bessel Functions, etc.

1 and v -- ~, then we have the L~PLAOE transform of the trignometrie function

sin I2 a:t~11] namely 1 )~ 1 ),] 1 1/~1 ~'~--1 (44) 2:+~(~}~p -~- ~. (-- 1)~n-'~ ~-~] ~=o

[ ~), 1 X 1 ~, 4a"e:~"~: i ~F~,,_~ 1+~+_n ' + ~ n ' n""p ~ q- ...... , a n; n n where

1 and n is any positive integer. Seanilarly by taking in (3) v----- we get 4' the L&PL&CE trasform of

cos [2a:t ~1] namely

(45) p~=o ),I r(~+ ),)

~+7~,1 X l+n;)" 4a"e~="~ ) 1 +: ..... ~_ ~-: ~- where R(p)> 2 I /;m/"(a-~}l and n is any positive integer. Formulae (44) and \ / (45) are new theorems of interest because they are widely utilized for special values of n (n--1, 2, 3, ...) in l~athematical Physics. Finally it may be noted that many new particular cases are obtained for n--1, 2, 3, .... in (4). F. iV[. RAGAB: The Laplace Transform of tl~e Bessel Functions, etc. 335

REFERENCES

[1] :ERDELYI~A, MAGNUS, W., OBERttETT][NGER F., and T~Ico~I F., Tables of integral transforms, VoL (I) Mcgraw 4ilt. New York (1954). [2] EaDELYI, A. and COSSAR S., Dictiona~'y of Laplace transfo~'m., Adm. Comp. Ser. London (1946}. [3] MACt~OBERT, ~. M. Functions of a Compleoc Variable, 4 the edit.) Macmillan and Co. (1954}. [4] MACROBERT, T. ])f. ,, Proc. Glasg. Math. Assoc.* ~rol. (3) Integrals allied to Airy's inte- grals, pp. 9i.93, (1957). [5] :ERDELYI .A_., MAGNUS W, ...~ Higher Transcendental functions, Yel. (I) Mcgraw ]~il], ~ew York (1954). [6] t~AGAB, F. M., Integrals involwing products of Bessel Functions, ~Arnali di Maiematica pura ed applicate~ Vol. LVI. pp. 301-312, (1961). [7] ~AOROBERT~ T.M., Evaluation of an E.function ~vhen t,vo of the upper parameters differ by an integer, ,Proc. G]asg. Mat. Assoc. ),, Vol. (V} pp. 30-3~, {1961). [8] ----, Multiplication formulae for the E.functivns rega~'ded as functions of their para. meters, ~Pacific. journ, cf Math., Vol. 9, pp. 759.761 {1959).